# Links of singular points of complex hypersurfaces

## 1 Introduction

The links of singular points of complex hypersurfaces provide a large class of examples of highly-connected odd dimensional manifolds. A standard reference is [Milnor1968]. See also [Hirzebruch&Mayer1968] and [Dimca1992].

These manifolds are the boundaries of highly-connected, stably parallelisable even dimensional manifolds and hence stably parallelisable themselves. In the case of singular points of complex curves, the link of such a singular point is a fibered link in the $3$${{Stub}} == Introduction == ; The links of singular points of complex hypersurfaces provide a large class of examples of highly-connected odd dimensional manifolds. A standard reference is {{cite|Milnor1968}}. See also {{cite|Hirzebruch&Mayer1968}} and {{cite|Dimca1992}}. These manifolds are the boundaries of [[highly-connected]], [[stably parallelisable]] even dimensional manifolds and hence stably parallelisable themselves. In the case of singular points of complex curves, the link of such a singular point is a [[fibered link]] in the -dimensional sphere S^3. == Construction and properties== ; Let f(z_1, \dots, z_{n+1}) be a non-constant polynomial in n+1 complex variables. A complex hypersurface V is the algebraic set consisting of points z=(z_1, \dots, z_{n+1}) such that f(z)=0. A regular point z \in V is a point at which some partial derivative \partial f /\partial z_j does not vanish; if at a point z \in V all the partial derivatives \partial f / \partial z_j vanish, z is called a '''singular point''' of V. Near a regular point z, the complex hypersurface V is a smooth manifold of real dimension n; in a small neighborhood of a singular point z, the topology of the complex hypersurface V is more complicated. One way to look at the topology near z, due to Brauner, is to look at the intersection of V with a (2n+1)-dimensioanl sphere of small radius \epsilon S_{\epsilon} centered at z. \begin{thm} The space K=V\cap S_{\epsilon} is (n-2)-connected. \end{thm} The homeomorphism type of K is independent of the small parameter \epsilon, it is called the '''link''' of the singular point z. \begin{thm}(Fibration Theorem)\label{fibration} For \epsilon sufficiently small, the space S_{\epsilon}-K is a smooth fiber bundle over S^1, with projection map \phi \colon S_{\epsilon}-K \to S^1, z \mapsto f(z)/|f(z)|. Each fiber F_{\theta} is parallelizable and has the homotopy type of a finite CW-complex of dimension n. \end{thm} The fiber F_{\theta} is usually called the '''Milnor fiber''' of the singular point z. A singular point z is '''isolated''' if there is no other singular point in some small neighborhood of z. In this special situation, the above theorems are strengthened to the following \begin{thm} Each fiber F_{\theta} is a smooth parallelizable manifold, the closure \overline{F_{\theta}} has boundary K and the homotopy type of a bouquet of n-spheres S^n\vee \cdots \vee S^n. \end{thm} == Invariants == ; Seen from the above section, the link K of an isolated singular point z of a complex hypersurface V of complex dimension n is an (n-2)-connected (2n-1)-dimensional closed smooth manifold. In high dimensional topology, these are called [[highly-connected|highly connected manifolds]], since for a (2n-1)-dimensional closed manifold M which is not a homotopy sphere, (n-2) is the highest connectivity M could have. Therefore to understand the classification and invariants of the links K one needs to understand the classification and invariants of [[highly-connected]] odd dimensional manifolds. On the other hand, as the link K is closely related to the singular point z of the complex hypersurfaces, some of the topological invariants of K are computable from the polynomial. Let V be a complex hypersurface defined by f(z_1, \dots, z_{n+1}), z^0 be an isolated singular point of V. Let g_j=\partial f /\partial z_j j=1, \dots, n+1. By putting all these g_j's together we get the gradient field of f, which can be viewed as a map g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}, z \mapsto (g_1(z), \dots, g_{n+1}(z)). If z^0 is an isolated singular point, then z \mapsto g(z)/||g(z)|| is a well-defined map from a small sphere S_{\epsilon} centered at z^0 to the unit sphere S^{2n+1} of \mathbb C^{n+1}. The mapping degree \mu is called the multiplicity of the isolated singular point z^0. (\mu is also called the Milnor number of z^0.) \begin{thm} The middle homology group H_n(F_{\theta}) is a free abelian group of rank \mu. \end{thm} Furthermore, the homology groups of the link K are determined from the long homology exact sequence \cdots \to H_n(\overline{F_{\theta}}) \stackrel{j_*}{\rightarrow} H_n(\overline{F_{\theta}}, K) \stackrel{\partial}{\rightarrow} \widetilde{H}_{n-1}(K) \to 0 of the pair (\overline{F_{\theta}},K). The map j_* is the adjoint of the intersection pairing on \overline{F_{\theta}} s \colon H_n(\overline{F_{\theta}}) \otimes H_n(\overline{F_{\theta}}) \to \mathbb Z. A symmetric or skew symmetric bilinear form on a free abelian group is usually referred to as a lattice. s is called the '''Milnor lattice''' of the singular point. Thus the homology groups of the link K is completely determined by the Milnor lattice of the singular point. == Topological spheres as links of singular points == ; Especially, the link K is an integral [[homology sphere]] if and only if the intersection form s is unimodular, i.~e.~the matrix of s has determinant \pm1. If n\ge 3, the [[Generalized Poincare Conjecture]] implies that K is a topological sphere. By Theorem \ref{fibration}, there is a smooth fiber bundle over S^1 with fiber F_{\theta}. The natural action of a generator of \pi_1(S^1) induces the characteristic homeomorphism h of the fiber F_0=\phi^{-1}. Let h_* \colon H_n(F_0) \to H_n(F_0) be the induced isomorphism on homology and \Delta(t)=\det(tI-h_*) be the '''characteristic polynomial''' of the linear transformation h_*. It's a consequence of the [[Wang sequence]] associated with the fiber bundle over S^1 that \begin{lemma} For n \ne 2 the manifolds K is a topological sphere is and only if the integer \Delta(1)=\det(I-h_*) equals to \pm 1. \end{lemma} When K is a topological sphere, as it is the boundary of an (n-1)-connected parallelisable n-manifold \overline{F_0}, our knowledge of [[exotic spheres]] allows us to determine the diffeomorphism class of K completely: * if n is even, the diffeomorphism class of K is determined by the [[Intersection forms#Algebraic invariants|signature]] of the intersection pairing s \colon H_n(\overline{F_0}) \otimes H_n(\overline{F_0}) \to \mathbb Z * if n is odd, the diffeomorphism class of K is determined by the [[wikipedia:Kervaire invariant|Kervaire invariant]] c(\overline{F_0}) \in \mathbb Z_2 which was computed in {{cite|Levine1966}} c(\overline{F_0})=0 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 1 \pmod 8 c(\overline{F_0})=1 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 3 \pmod 8 == Examples == ; The Brieskorn singularitites is a class of singular points which have been extensively studied. These are defined by the polynomials of the form f(z_1, \dots, z_{n+1})=(z_1)^{a_1}+\dots +(z_{n+1})^{a_{n+1}} where a_1, \dots, a_{n+1} are integers \ge 2. The origin is an isolated singular point of f. \begin{thm}(Brieskorn-Pham) The group H_n(F_{\theta}) is free abelian of rank \mu = (a_1-1)(a_2-1)\cdots (a_{n+1}-1.) The characteristic polynomial is \Delta(t)=\prod(t-\omega_1 \omega_2 \cdots \omega_{n+1}), where each \omega_j ranges over all a_j-th root of unit other than 3$-dimensional sphere $S^3$$S^3$.

## 2 Construction and properties

Let $f(z_1, \dots, z_{n+1})$$f(z_1, \dots, z_{n+1})$ be a non-constant polynomial in $n+1$$n+1$ complex variables. A complex hypersurface $V$$V$ is the algebraic set consisting of points $z=(z_1, \dots, z_{n+1})$$z=(z_1, \dots, z_{n+1})$ such that $f(z)=0$$f(z)=0$. A regular point $z \in V$$z \in V$ is a point at which some partial derivative $\partial f /\partial z_j$$\partial f /\partial z_j$ does not vanish; if at a point $z \in V$$z \in V$ all the partial derivatives $\partial f / \partial z_j$$\partial f / \partial z_j$ vanish, $z$$z$ is called a singular point of $V$$V$.

Near a regular point $z$$z$, the complex hypersurface $V$$V$ is a smooth manifold of real dimension $2n$$2n$; in a small neighborhood of a singular point $z$$z$, the topology of the complex hypersurface $V$$V$ is more complicated. One way to look at the topology near $z$$z$, due to Brauner, is to look at the intersection of $V$$V$ with a $(2n+1)$$(2n+1)$-dimensioanl sphere of small radius $\epsilon$$\epsilon$ $S_{\epsilon}$$S_{\epsilon}$ centered at $z$$z$.

\begin{thm} The space $K=V\cap S_{\epsilon}$$K=V\cap S_{\epsilon}$ is $(n-2)$$(n-2)$-connected. \end{thm}

The homeomorphism type of $K$$K$ is independent of the small parameter $\epsilon$$\epsilon$, it is called the link of the singular point $z$$z$.

\begin{thm}(Fibration Theorem) For $\epsilon$$\epsilon$ sufficiently small, the space $S_{\epsilon}-K$$S_{\epsilon}-K$ is a smooth fiber bundle over $S^1$$S^1$, with projection map $\phi \colon S_{\epsilon}-K \to S^1$$\phi \colon S_{\epsilon}-K \to S^1$, $z \mapsto f(z)/|f(z)|$$z \mapsto f(z)/|f(z)|$. Each fiber $F_{\theta}$$F_{\theta}$ is parallelizable and has the homotopy type of a finite CW-complex of dimension $n$$n$. \end{thm}

The fiber $F_{\theta}$$F_{\theta}$ is usually called the Milnor fiber of the singular point $z$$z$.

A singular point $z$$z$ is isolated if there is no other singular point in some small neighborhood of $z$$z$.

In this special situation, the above theorems are strengthened to the following

\begin{thm} Each fiber $F_{\theta}$$F_{\theta}$ is a smooth parallelizable manifold, the closure $\overline{F_{\theta}}$$\overline{F_{\theta}}$ has boundary $K$$K$ and the homotopy type of a bouquet of $n$$n$-spheres $S^n\vee \cdots \vee S^n$$S^n\vee \cdots \vee S^n$. \end{thm}

## 3 Invariants

Seen from the above section, the link $K$$K$ of an isolated singular point $z$$z$ of a complex hypersurface $V$$V$ of complex dimension $n$$n$ is an $(n-2)$$(n-2)$-connected $(2n-1)$$(2n-1)$-dimensional closed smooth manifold. In high dimensional topology, these are called highly connected manifolds, since for a $(2n-1)$$(2n-1)$-dimensional closed manifold $M$$M$ which is not a homotopy sphere, $(n-2)$$(n-2)$ is the highest connectivity $M$$M$ could have. Therefore to understand the classification and invariants of the links $K$$K$ one needs to understand the classification and invariants of highly-connected odd dimensional manifolds.

On the other hand, as the link $K$$K$ is closely related to the singular point $z$$z$ of the complex hypersurfaces, some of the topological invariants of $K$$K$ are computable from the polynomial.

Let $V$$V$ be a complex hypersurface defined by $f(z_1, \dots, z_{n+1})$$f(z_1, \dots, z_{n+1})$, $z^0$$z^0$ be an isolated singular point of $V$$V$. Let $g_j=\partial f /\partial z_j$$g_j=\partial f /\partial z_j$ $j=1, \dots, n+1$$j=1, \dots, n+1$. By putting all these $g_j$$g_j$'s together we get the gradient field of $f$$f$, which can be viewed as a map $g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}$$g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}$, $z \mapsto (g_1(z), \dots, g_{n+1}(z))$$z \mapsto (g_1(z), \dots, g_{n+1}(z))$. If $z^0$$z^0$ is an isolated singular point, then $z \mapsto g(z)/||g(z)||$$z \mapsto g(z)/||g(z)||$ is a well-defined map from a small sphere $S_{\epsilon}$$S_{\epsilon}$ centered at $z^0$$z^0$ to the unit sphere $S^{2n+1}$$S^{2n+1}$ of $\mathbb C^{n+1}$$\mathbb C^{n+1}$. The mapping degree $\mu$$\mu$ is called the multiplicity of the isolated singular point $z^0$$z^0$. ($\mu$$\mu$ is also called the Milnor number of $z^0$$z^0$.)

\begin{thm} The middle homology group $H_n(F_{\theta})$$H_n(F_{\theta})$ is a free abelian group of rank $\mu$$\mu$. \end{thm}

Furthermore, the homology groups of the link $K$$K$ are determined from the long homology exact sequence

$\displaystyle \cdots \to H_n(\overline{F_{\theta}}) \stackrel{j_*}{\rightarrow} H_n(\overline{F_{\theta}}, K) \stackrel{\partial}{\rightarrow} \widetilde{H}_{n-1}(K) \to 0$

of the pair $(\overline{F_{\theta}},K)$$(\overline{F_{\theta}},K)$. The map $j_*$$j_*$ is the adjoint of the intersection pairing on $\overline{F_{\theta}}$$\overline{F_{\theta}}$

$\displaystyle s \colon H_n(\overline{F_{\theta}}) \otimes H_n(\overline{F_{\theta}}) \to \mathbb Z.$

A symmetric or skew symmetric bilinear form on a free abelian group is usually referred to as a lattice. $s$$s$ is called the Milnor lattice of the singular point. Thus the homology groups of the link $K$$K$ is completely determined by the Milnor lattice of the singular point.

## Topological spheres as links of singular points

Especially, the link $K$$K$ is an integral homology sphere if and only if the intersection form $s$$s$ is unimodular, i.~e.~the matrix of $s$$s$ has determinant $\pm1$$\pm1$. If $n\ge 3$$n\ge 3$, the Generalized Poincare Conjecture implies that $K$$K$ is a topological sphere.

By Theorem 2, there is a smooth fiber bundle over $S^1$$S^1$ with fiber $F_{\theta}$$F_{\theta}$. The natural action of a generator of $\pi_1(S^1)$$\pi_1(S^1)$ induces the characteristic homeomorphism $h$$h$ of the fiber $F_0=\phi^{-1}$$F_0=\phi^{-1}$. Let $h_* \colon H_n(F_0) \to H_n(F_0)$$h_* \colon H_n(F_0) \to H_n(F_0)$ be the induced isomorphism on homology and $\Delta(t)=\det(tI-h_*)$$\Delta(t)=\det(tI-h_*)$ be the characteristic polynomial of the linear transformation $h_*$$h_*$. It's a consequence of the Wang sequence associated with the fiber bundle over $S^1$$S^1$ that

Lemma 5.1. For $n \ne 2$$n \ne 2$ the manifolds $K$$K$ is a topological sphere is and only if the integer $\Delta(1)=\det(I-h_*)$$\Delta(1)=\det(I-h_*)$ equals to $\pm 1$$\pm 1$.

When $K$$K$ is a topological sphere, as it is the boundary of an $(n-1)$$(n-1)$-connected parallelisable $2n$$2n$-manifold $\overline{F_0}$$\overline{F_0}$, our knowledge of exotic spheres allows us to determine the diffeomorphism class of $K$$K$ completely:

• if $n$$n$ is even, the diffeomorphism class of $K$$K$ is determined by the signature of the intersection pairing
$\displaystyle s \colon H_n(\overline{F_0}) \otimes H_n(\overline{F_0}) \to \mathbb Z$
• if $n$$n$ is odd, the diffeomorphism class of $K$$K$ is determined by the Kervaire invariant
$\displaystyle c(\overline{F_0}) \in \mathbb Z_2$

which was computed in [Levine1966]

$\displaystyle c(\overline{F_0})=0 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 1 \pmod 8$
$\displaystyle c(\overline{F_0})=1 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 3 \pmod 8$

## 4 Examples

The Brieskorn singularitites is a class of singular points which have been extensively studied. These are defined by the polynomials of the form

$\displaystyle f(z_1, \dots, z_{n+1})=(z_1)^{a_1}+\dots +(z_{n+1})^{a_{n+1}}$

where $a_1, \dots, a_{n+1}$$a_1, \dots, a_{n+1}$ are integers $\ge 2$$\ge 2$. The origin is an isolated singular point of $f$$f$.

\begin{thm}(Brieskorn-Pham) The group $H_n(F_{\theta})$$H_n(F_{\theta})$ is free abelian of rank

$\displaystyle \mu = (a_1-1)(a_2-1)\cdots (a_{n+1}-1.)$

The characteristic polynomial is

$\displaystyle \Delta(t)=\prod(t-\omega_1 \omega_2 \cdots \omega_{n+1}),$

where each $\omega_j$$\omega_j$ ranges over all $a_j$$a_j$-th root of unit other than $1$$1$. \end{thm}

The link $K$$K$ is called a Brieskorn variety.

For $a_1=\cdots=a_{n+1}=2$$a_1=\cdots=a_{n+1}=2$, it's seen from the defining equations that the link $K$$K$ is the sphere bundle of the tangent bundle of the $n$$n$-sphere, i.~e.~the Stiefel manifold $V_{n+1,2}(\mathbb R)$$V_{n+1,2}(\mathbb R)$.

The simplest nontrivial example is $a_1=\cdots =a_n=2$$a_1=\cdots =a_n=2$, $a_{n+1}=3$$a_{n+1}=3$. Then $\omega_1 = \cdots \omega_n=-1$$\omega_1 = \cdots \omega_n=-1$, $\omega_{n+1}=(-1\pm \sqrt{3})/2$$\omega_{n+1}=(-1\pm \sqrt{3})/2$. The characteristic polynomial is

$\displaystyle \Delta(t)=t^2-t+1 \ \ \textrm{for} \ \ n \ \ \textrm{odd}$
$\displaystyle \Delta(t)=t^2+t+1 \ \ \textrm{for} \ \ n \ \ \textrm{even}.$

For $n=2k+1$$n=2k+1$ we have $\Delta(1)=1$$\Delta(1)=1$ so the link $K$$K$ is a topological sphere of dimension $4k+1$$4k+1$; $\Delta(-1)=3$$\Delta(-1)=3$, thus by [Levine1966] $K$$K$ has nontrivial Kervaire invariant. Especially for $k=2$$k=2$ $K^9$$K^9$ is the Kervaire sphere.

The above example is a special case of the $A_k$$A_k$-singularities, whose defining polynomial is

$\displaystyle f(z_1, \dots, z_{n+1})=z_1^2+\cdots+z_n^2+z_{n+1}^{k+1}$

$k$$k$ being an integer $\ge 1$$\ge 1$. The Milnor lattice of an $A_k$$A_k$-singularity is represented by the Dynkin diagram of the simple Lie algebra $A_k$$A_k$. When $n=3$$n=3$, the diffeomorphism classification of the link $K$$K$ is obtained from its Milnor lattice and the classification of simply-connected 5-manifolds:

• $K$$K$ is diffeomorphic to $S^2 \times S^3$$S^2 \times S^3$ if $k$$k$ is odd;
• $K$$K$ is diffeomorphic to $S^5$$S^5$ if $k$$k$ is even.

In this dimension, the diffeomorphism classification of the link $K$$K$ of other types of singular points can be obtained in the same way, once we know the Milnor lattice of the singular point.

## 5 Further discussion

The link $K$$K$ of a singular point $z$$z$ is the intersection of the hypersurface $V$$V$ defined by $f$$f$ and the sphere $S_{\epsilon}$$S_{\epsilon}$ in the ambient space $\mathbb C^{n+1}$$\mathbb C^{n+1}$. Therefore it inherits certain symmetries from that of the singular point. As an example, consider an $A_k$$A_k$-singularity in $\mathbb C^4$$\mathbb C^4$. There is an orientation preserving involution

$\displaystyle \tau_k \colon \mathbb C^4 \to \mathbb C^4, \ \ (z_1, z_2, _3, z_4) \mapsto (-z_1, -z_2, -z_3, z_4).$

$\tau_k$$\tau_k$ induces an orientation preserving free involution of $K\cong S^5$$K\cong S^5$ or $S^2 \times S^3$$S^2 \times S^3$. For $k=0,2,4,6$$k=0,2,4,6$, $\tau_k$$\tau_k$'s provide all the 4 smooth free involutions on $S^5$$S^5$ (see [Geiges&Thomas1998]).

## 6 References

$. \end{thm} The link$K$is called a [[Exotic spheres#Brieskorn varieties|Brieskorn variety]]. For$a_1=\cdots=a_{n+1}=2$, it's seen from the defining equations that the link$K$is the sphere bundle of the tangent bundle of the$n$-sphere, i.~e.~the [[wikipedia:Stiefel manifold|Stiefel manifold]]$V_{n+1,2}(\mathbb R)$. The simplest nontrivial example is$a_1=\cdots =a_n=2$,$a_{n+1}=3$. Then$\omega_1 = \cdots \omega_n=-1$,$\omega_{n+1}=(-1\pm \sqrt{3})/2$. The characteristic polynomial is $$\Delta(t)=t^2-t+1 \ \ \textrm{for} \ \ n \ \ \textrm{odd}$$ $$\Delta(t)=t^2+t+1 \ \ \textrm{for} \ \ n \ \ \textrm{even}.$$ For$n=2k+1$we have$\Delta(1)=1$so the link$K$is a topological sphere of dimension k+1$; $\Delta(-1)=3$, thus by {{cite|Levine1966}} $K$ has nontrivial [[wikipedia:Kervaire invariant|Kervaire invariant]]. Especially for $k=2$ $K^9$ is the [[Exotic spheres#Plumbing|Kervaire sphere]]. The above example is a special case of the $A_k$-singularities, whose defining polynomial is $$f(z_1, \dots, z_{n+1})=z_1^2+\cdots+z_n^2+z_{n+1}^{k+1}$$ $k$ being an integer $\ge 1$. The Milnor lattice of an $A_k$-singularity is represented by the Dynkin diagram of the simple Lie algebra $A_k$. When $n=3$, the diffeomorphism classification of the link $K$ is obtained from its Milnor lattice and the classification of [[5-manifolds: 1-connected|simply-connected 5-manifolds]]: * $K$ is diffeomorphic to $S^2 \times S^3$ if $k$ is odd; * $K$ is diffeomorphic to $S^5$ if $k$ is even. In this dimension, the diffeomorphism classification of the link $K$ of other types of singular points can be obtained in the same way, once we know the [[Links of singular points of complex hypersurfaces#Invariants|Milnor lattice]] of the singular point. == Further discussion == ; The link $K$ of a singular point $z$ is the intersection of the hypersurface $V$ defined by $f$ and the sphere $S_{\epsilon}$ in the ambient space $\mathbb C^{n+1}$. Therefore it inherits certain symmetries from that of the singular point. As an example, consider an $A_k$-singularity in $\mathbb C^4$. There is an orientation preserving involution $$\tau_k \colon \mathbb C^4 \to \mathbb C^4, \ \ (z_1, z_2, _3, z_4) \mapsto (-z_1, -z_2, -z_3, z_4).$$ $\tau_k$ induces an orientation preserving free involution of $K\cong S^5$ or $S^2 \times S^3$. For $k=0,2,4,6$, $\tau_k$'s provide all the 4 smooth free involutions on $S^5$ (see {{cite|Geiges&Thomas1998}}). == References == {{#RefList:}} [[Category:Manifolds]]3-dimensional sphere $S^3$$S^3$.

## 2 Construction and properties

Let $f(z_1, \dots, z_{n+1})$$f(z_1, \dots, z_{n+1})$ be a non-constant polynomial in $n+1$$n+1$ complex variables. A complex hypersurface $V$$V$ is the algebraic set consisting of points $z=(z_1, \dots, z_{n+1})$$z=(z_1, \dots, z_{n+1})$ such that $f(z)=0$$f(z)=0$. A regular point $z \in V$$z \in V$ is a point at which some partial derivative $\partial f /\partial z_j$$\partial f /\partial z_j$ does not vanish; if at a point $z \in V$$z \in V$ all the partial derivatives $\partial f / \partial z_j$$\partial f / \partial z_j$ vanish, $z$$z$ is called a singular point of $V$$V$.

Near a regular point $z$$z$, the complex hypersurface $V$$V$ is a smooth manifold of real dimension $2n$$2n$; in a small neighborhood of a singular point $z$$z$, the topology of the complex hypersurface $V$$V$ is more complicated. One way to look at the topology near $z$$z$, due to Brauner, is to look at the intersection of $V$$V$ with a $(2n+1)$$(2n+1)$-dimensioanl sphere of small radius $\epsilon$$\epsilon$ $S_{\epsilon}$$S_{\epsilon}$ centered at $z$$z$.

\begin{thm} The space $K=V\cap S_{\epsilon}$$K=V\cap S_{\epsilon}$ is $(n-2)$$(n-2)$-connected. \end{thm}

The homeomorphism type of $K$$K$ is independent of the small parameter $\epsilon$$\epsilon$, it is called the link of the singular point $z$$z$.

\begin{thm}(Fibration Theorem) For $\epsilon$$\epsilon$ sufficiently small, the space $S_{\epsilon}-K$$S_{\epsilon}-K$ is a smooth fiber bundle over $S^1$$S^1$, with projection map $\phi \colon S_{\epsilon}-K \to S^1$$\phi \colon S_{\epsilon}-K \to S^1$, $z \mapsto f(z)/|f(z)|$$z \mapsto f(z)/|f(z)|$. Each fiber $F_{\theta}$$F_{\theta}$ is parallelizable and has the homotopy type of a finite CW-complex of dimension $n$$n$. \end{thm}

The fiber $F_{\theta}$$F_{\theta}$ is usually called the Milnor fiber of the singular point $z$$z$.

A singular point $z$$z$ is isolated if there is no other singular point in some small neighborhood of $z$$z$.

In this special situation, the above theorems are strengthened to the following

\begin{thm} Each fiber $F_{\theta}$$F_{\theta}$ is a smooth parallelizable manifold, the closure $\overline{F_{\theta}}$$\overline{F_{\theta}}$ has boundary $K$$K$ and the homotopy type of a bouquet of $n$$n$-spheres $S^n\vee \cdots \vee S^n$$S^n\vee \cdots \vee S^n$. \end{thm}

## 3 Invariants

Seen from the above section, the link $K$$K$ of an isolated singular point $z$$z$ of a complex hypersurface $V$$V$ of complex dimension $n$$n$ is an $(n-2)$$(n-2)$-connected $(2n-1)$$(2n-1)$-dimensional closed smooth manifold. In high dimensional topology, these are called highly connected manifolds, since for a $(2n-1)$$(2n-1)$-dimensional closed manifold $M$$M$ which is not a homotopy sphere, $(n-2)$$(n-2)$ is the highest connectivity $M$$M$ could have. Therefore to understand the classification and invariants of the links $K$$K$ one needs to understand the classification and invariants of highly-connected odd dimensional manifolds.

On the other hand, as the link $K$$K$ is closely related to the singular point $z$$z$ of the complex hypersurfaces, some of the topological invariants of $K$$K$ are computable from the polynomial.

Let $V$$V$ be a complex hypersurface defined by $f(z_1, \dots, z_{n+1})$$f(z_1, \dots, z_{n+1})$, $z^0$$z^0$ be an isolated singular point of $V$$V$. Let $g_j=\partial f /\partial z_j$$g_j=\partial f /\partial z_j$ $j=1, \dots, n+1$$j=1, \dots, n+1$. By putting all these $g_j$$g_j$'s together we get the gradient field of $f$$f$, which can be viewed as a map $g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}$$g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}$, $z \mapsto (g_1(z), \dots, g_{n+1}(z))$$z \mapsto (g_1(z), \dots, g_{n+1}(z))$. If $z^0$$z^0$ is an isolated singular point, then $z \mapsto g(z)/||g(z)||$$z \mapsto g(z)/||g(z)||$ is a well-defined map from a small sphere $S_{\epsilon}$$S_{\epsilon}$ centered at $z^0$$z^0$ to the unit sphere $S^{2n+1}$$S^{2n+1}$ of $\mathbb C^{n+1}$$\mathbb C^{n+1}$. The mapping degree $\mu$$\mu$ is called the multiplicity of the isolated singular point $z^0$$z^0$. ($\mu$$\mu$ is also called the Milnor number of $z^0$$z^0$.)

\begin{thm} The middle homology group $H_n(F_{\theta})$$H_n(F_{\theta})$ is a free abelian group of rank $\mu$$\mu$. \end{thm}

Furthermore, the homology groups of the link $K$$K$ are determined from the long homology exact sequence

$\displaystyle \cdots \to H_n(\overline{F_{\theta}}) \stackrel{j_*}{\rightarrow} H_n(\overline{F_{\theta}}, K) \stackrel{\partial}{\rightarrow} \widetilde{H}_{n-1}(K) \to 0$

of the pair $(\overline{F_{\theta}},K)$$(\overline{F_{\theta}},K)$. The map $j_*$$j_*$ is the adjoint of the intersection pairing on $\overline{F_{\theta}}$$\overline{F_{\theta}}$

$\displaystyle s \colon H_n(\overline{F_{\theta}}) \otimes H_n(\overline{F_{\theta}}) \to \mathbb Z.$

A symmetric or skew symmetric bilinear form on a free abelian group is usually referred to as a lattice. $s$$s$ is called the Milnor lattice of the singular point. Thus the homology groups of the link $K$$K$ is completely determined by the Milnor lattice of the singular point.

## Topological spheres as links of singular points

Especially, the link $K$$K$ is an integral homology sphere if and only if the intersection form $s$$s$ is unimodular, i.~e.~the matrix of $s$$s$ has determinant $\pm1$$\pm1$. If $n\ge 3$$n\ge 3$, the Generalized Poincare Conjecture implies that $K$$K$ is a topological sphere.

By Theorem 2, there is a smooth fiber bundle over $S^1$$S^1$ with fiber $F_{\theta}$$F_{\theta}$. The natural action of a generator of $\pi_1(S^1)$$\pi_1(S^1)$ induces the characteristic homeomorphism $h$$h$ of the fiber $F_0=\phi^{-1}$$F_0=\phi^{-1}$. Let $h_* \colon H_n(F_0) \to H_n(F_0)$$h_* \colon H_n(F_0) \to H_n(F_0)$ be the induced isomorphism on homology and $\Delta(t)=\det(tI-h_*)$$\Delta(t)=\det(tI-h_*)$ be the characteristic polynomial of the linear transformation $h_*$$h_*$. It's a consequence of the Wang sequence associated with the fiber bundle over $S^1$$S^1$ that

Lemma 5.1. For $n \ne 2$$n \ne 2$ the manifolds $K$$K$ is a topological sphere is and only if the integer $\Delta(1)=\det(I-h_*)$$\Delta(1)=\det(I-h_*)$ equals to $\pm 1$$\pm 1$.

When $K$$K$ is a topological sphere, as it is the boundary of an $(n-1)$$(n-1)$-connected parallelisable $2n$$2n$-manifold $\overline{F_0}$$\overline{F_0}$, our knowledge of exotic spheres allows us to determine the diffeomorphism class of $K$$K$ completely:

• if $n$$n$ is even, the diffeomorphism class of $K$$K$ is determined by the signature of the intersection pairing
$\displaystyle s \colon H_n(\overline{F_0}) \otimes H_n(\overline{F_0}) \to \mathbb Z$
• if $n$$n$ is odd, the diffeomorphism class of $K$$K$ is determined by the Kervaire invariant
$\displaystyle c(\overline{F_0}) \in \mathbb Z_2$

which was computed in [Levine1966]

$\displaystyle c(\overline{F_0})=0 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 1 \pmod 8$
$\displaystyle c(\overline{F_0})=1 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 3 \pmod 8$

## 4 Examples

The Brieskorn singularitites is a class of singular points which have been extensively studied. These are defined by the polynomials of the form

$\displaystyle f(z_1, \dots, z_{n+1})=(z_1)^{a_1}+\dots +(z_{n+1})^{a_{n+1}}$

where $a_1, \dots, a_{n+1}$$a_1, \dots, a_{n+1}$ are integers $\ge 2$$\ge 2$. The origin is an isolated singular point of $f$$f$.

\begin{thm}(Brieskorn-Pham) The group $H_n(F_{\theta})$$H_n(F_{\theta})$ is free abelian of rank

$\displaystyle \mu = (a_1-1)(a_2-1)\cdots (a_{n+1}-1.)$

The characteristic polynomial is

$\displaystyle \Delta(t)=\prod(t-\omega_1 \omega_2 \cdots \omega_{n+1}),$

where each $\omega_j$$\omega_j$ ranges over all $a_j$$a_j$-th root of unit other than $1$$1$. \end{thm}

The link $K$$K$ is called a Brieskorn variety.

For $a_1=\cdots=a_{n+1}=2$$a_1=\cdots=a_{n+1}=2$, it's seen from the defining equations that the link $K$$K$ is the sphere bundle of the tangent bundle of the $n$$n$-sphere, i.~e.~the Stiefel manifold $V_{n+1,2}(\mathbb R)$$V_{n+1,2}(\mathbb R)$.

The simplest nontrivial example is $a_1=\cdots =a_n=2$$a_1=\cdots =a_n=2$, $a_{n+1}=3$$a_{n+1}=3$. Then $\omega_1 = \cdots \omega_n=-1$$\omega_1 = \cdots \omega_n=-1$, $\omega_{n+1}=(-1\pm \sqrt{3})/2$$\omega_{n+1}=(-1\pm \sqrt{3})/2$. The characteristic polynomial is

$\displaystyle \Delta(t)=t^2-t+1 \ \ \textrm{for} \ \ n \ \ \textrm{odd}$
$\displaystyle \Delta(t)=t^2+t+1 \ \ \textrm{for} \ \ n \ \ \textrm{even}.$

For $n=2k+1$$n=2k+1$ we have $\Delta(1)=1$$\Delta(1)=1$ so the link $K$$K$ is a topological sphere of dimension $4k+1$$4k+1$; $\Delta(-1)=3$$\Delta(-1)=3$, thus by [Levine1966] $K$$K$ has nontrivial Kervaire invariant. Especially for $k=2$$k=2$ $K^9$$K^9$ is the Kervaire sphere.

The above example is a special case of the $A_k$$A_k$-singularities, whose defining polynomial is

$\displaystyle f(z_1, \dots, z_{n+1})=z_1^2+\cdots+z_n^2+z_{n+1}^{k+1}$

$k$$k$ being an integer $\ge 1$$\ge 1$. The Milnor lattice of an $A_k$$A_k$-singularity is represented by the Dynkin diagram of the simple Lie algebra $A_k$$A_k$. When $n=3$$n=3$, the diffeomorphism classification of the link $K$$K$ is obtained from its Milnor lattice and the classification of simply-connected 5-manifolds:

• $K$$K$ is diffeomorphic to $S^2 \times S^3$$S^2 \times S^3$ if $k$$k$ is odd;
• $K$$K$ is diffeomorphic to $S^5$$S^5$ if $k$$k$ is even.

In this dimension, the diffeomorphism classification of the link $K$$K$ of other types of singular points can be obtained in the same way, once we know the Milnor lattice of the singular point.

## 5 Further discussion

The link $K$$K$ of a singular point $z$$z$ is the intersection of the hypersurface $V$$V$ defined by $f$$f$ and the sphere $S_{\epsilon}$$S_{\epsilon}$ in the ambient space $\mathbb C^{n+1}$$\mathbb C^{n+1}$. Therefore it inherits certain symmetries from that of the singular point. As an example, consider an $A_k$$A_k$-singularity in $\mathbb C^4$$\mathbb C^4$. There is an orientation preserving involution

$\displaystyle \tau_k \colon \mathbb C^4 \to \mathbb C^4, \ \ (z_1, z_2, _3, z_4) \mapsto (-z_1, -z_2, -z_3, z_4).$

$\tau_k$$\tau_k$ induces an orientation preserving free involution of $K\cong S^5$$K\cong S^5$ or $S^2 \times S^3$$S^2 \times S^3$. For $k=0,2,4,6$$k=0,2,4,6$, $\tau_k$$\tau_k$'s provide all the 4 smooth free involutions on $S^5$$S^5$ (see [Geiges&Thomas1998]).